Brezina, Marian; Vaněk, Petr; Vassilevski, Panayot S. An improved convergence analysis of smoothed aggregation algebraic multigrid. (English) Zbl 1274.65315 Numer. Linear Algebra Appl. 19, No. 3, 441-469 (2012). Summary: We present an improved analysis of the smoothed aggregation algebraic multigrid method extending the original proof in the work of P. Vaněk et al. [Numer. Math. 88, No. 3, 559–579 (2001; Zbl 0992.65139)] and its modification by P. S. Vassilevski [Multilevel block factorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. New York, NY: Springer (2008; Zbl 1170.65001)]. The new result imposes fewer restrictions on the aggregates that makes it easier to verify in practice. Also, we extend a result of P. Vaněk [Appl. Math., Praha 57, No. 1, 1–10 (2012; Zbl 1249.65272)] that allows us to use aggressive coarsening at all levels. This is due to the properties of the special polynomial smoother that we use and analyze. In particular, we obtain bounds in the multilevel convergence estimates that are independent of the coarsening ratio. Numerical illustration is also provided. Cited in 1 ReviewCited in 16 Documents MSC: 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Keywords:smoothed aggregation; algebraic multigrid; convergence; polynomial smoother; aggressive coarsening; numerical examples Citations:Zbl 0992.65139; Zbl 1170.65001; Zbl 1249.65272 Software:CSparse PDFBibTeX XMLCite \textit{M. Brezina} et al., Numer. Linear Algebra Appl. 19, No. 3, 441--469 (2012; Zbl 1274.65315) Full Text: DOI Link References: [1] Vaněk, Convergence of algebraic multigrid based on smoothed aggregation, Numerische Mathematik 88 pp 559– (2001) · doi:10.1007/s211-001-8015-y [2] Vassilevski, Multilevel Block Factorization Preconditioners. Matrix-Based Analysis and Algorithms for Solving Finite Element Equations (2008) · Zbl 1170.65001 [3] Vaněk, Two-grid method for elasticity on unstructured meshes, SIAM Journal on Scientific Computing 21 pp 900– (1999) · Zbl 0952.65099 · doi:10.1137/S1064827596297112 [4] Vaněk, Smoothed prolongation multigrid with rapid coarsening and massive smoothing, Applications of Mathematics (2011) [5] Křížková, Two-level preconditioner with small coarse grid appropriate for unstructured meshes, Numerical Linear Algebra with Applications 3 (4) pp 255– (1996) · Zbl 0906.65114 · doi:10.1002/(SICI)1099-1506(199607/08)3:4<255::AID-NLA77>3.0.CO;2-2 [6] Bornemann, The cascadic multigrid method for elliptic problems, Numerische Mathematik 75 pp 135– (1996) · Zbl 0873.65107 · doi:10.1007/s002110050234 [7] Brezina M Heberton C Mandel J Vaněk P http://www-math.cudenver.edu/jmandel/papers/1999 [8] Shaidurov, Multigrid Methods for Finite Elements (1995) · Zbl 0837.65118 · doi:10.1007/978-94-015-8527-9 [9] Xu, The method of alternating projections and the method of subspace corrections in hilbert space, Journal of the American Mathematics Society 15 pp 573– (2002) · Zbl 0999.47015 · doi:10.1090/S0894-0347-02-00398-3 [10] Toselli, Domain Decomposition Methods-Algorithms and Theory (2005) · Zbl 1069.65138 · doi:10.1007/b137868 [11] Vaněk, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing 56 (3) pp 179– (1996) · Zbl 0851.65087 · doi:10.1007/BF02238511 [12] Davis, Direct Methods for Sparse Linear Systems (2006) · Zbl 1119.65021 · doi:10.1137/1.9780898718881 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.